Valency-based structural properties of gamma-sheet of boron clusters

Boron cluster sheets are two-dimensional boron atom-based formations called borophene. They are similar to the two-dimensional sheet known as graphene, which is composed of carbon atoms arranged in a hexagonal lattice. The unique electrical, mechanical, and thermal properties of borophene make it a sought-after substance for a variety of uses, such as catalysis, energy storage, and electronics. There are two ways to manufacture borophene: chemical vapor deposition and molecular beam epitaxy. Vertex-edge valency-based topological descriptors are a great example of a molecular descriptor that provides information on the connection of atoms in a molecule. These descriptions are based on the notion that a node’s value in a molecular network is the sum of the valency of those atoms that are directly connected to that node. In this article, we discussed some novel vertex-edge (ve) and edge-vertex (ev) topological descriptors and found their formulations for the boron cluster or borophene sheets. Also, we show the numerical and graphical comparison of these descriptors in this article.


Introduction
A fundamental study in the structural theory of chemical graphs is a molecular structure diagram, in which atoms are seen as atoms and lines as chemical bonds.If there is a path connecting any two atoms in a diagram, it is said to be linked.A system is a linked map with a loop and no other lines separating any two atoms.The number of atoms is linked to a given node.The valency of v is indicated by the symbol d v .The open neighbourhood of node v is the collection of all the atoms that are close to it, and it may be represented by the symbol N(v).When the node v, indicated by N [v], was included, the open neighbourhood changed to the closed neighbourhood.
The total length of the most constrained path between two atoms is the distance across both.In physical science, the concepts of valence and valency are pretty closely related.For basic diagram invariants, see [1].The qualities and unstudied material's natural exercises are predicted by the relationship between the QSPR and QSAR.Topological descriptors and a few physicochemical features are used in such compounds for estimating the bioactivity of chemical substances [2][3][4][5].In a diagram of a chemical compound, a number is used to express a topological descriptor that may be used to depict the highlighted chemical molecule and predict its physiochemical characteristics.
Wiener devised the topological descriptor structure in 1947.He presented the Wiener descriptor and was given an approximation of the alkanes' point of boiling [6][7][8][9][10].More than 3000 topological indicators have been characterized so far, however, no one descriptor is enough to determine all physicochemical characteristics, yet these topological descriptors together can partly do this.After that the Randić descriptor was given by cite7 in 1975.One of the most significant, widely used, and applicable topological descriptors is the Randić descriptor.Graph invariance is the subject of several surveys, studies, and books [11][12][13][14][15][16].For further detail, we refer to see [17][18][19][20][21][22].
Authors of [23], first proposed two unique valency concepts, the ve-valency and the evvalency, and [24,25] made a contribution to the research on the "ve-valency" and the "evvalency".When current descriptors were combined with the newly developed valency-based descriptors, the outcomes were improved, as shown in [26][27][28].It is currently shown that the ve-valency Zagreb descriptor has a stronger approximation performance than the original Zagreb descriptor.

The ve-valency and ev-valency based topological descriptors
The definition of an edge's ev-valency was provided by the contributors of citation number [23], and it is the count of atoms in the union of the closed neighbourhoods of u and v, where e = uv 2 E which is denoted by d ev (e).The count of routes of various lines that are incident to any node from the closed neighbourhood of v is known as the ve-valency of the node v 2 V, indicated by the symbol d ve (v).For the sake of this essay, we will assume that G is a connected graph with e = uv 2 E(G) while v 2 V.In this section, we provide some fundamental notions of ve-valency and ev-valency topological descriptors, [23,43].Some ev-valency related topological descriptors are: the ev-valency Randić descriptor, ansd ev-valency Zagreb descriptor, and topological descriptors related to ev-valency are: The first ve-valency Zagreb α descriptor, vevalency sum-connectivity descriptor, ve-valency harmonic descriptor, ve-valency geometricarithmetic descriptor, ve-valency atom-bond connectivity descriptor, ve-valency Randić descriptor, second ve-valency Zagreb descriptor, and first ve-valency Zagreb β descriptor.All these descriptors for any graph G can be found in [44], are formulated as:   They are similar to the two-dimensional sheet known as graphene, which is composed of carbon atoms arranged in a hexagonal lattice.The unique electrical, mechanical, and thermal properties of borophene make it a sought-after substance for a variety of uses, such as catalysis, energy storage, and electronics.There are two ways to manufacture borophene: chemical vapour deposition and molecular beam epitaxy.To create borophene sheets, many arrangements of boron atoms, such as triangular, honeycomb, and rectangular patterns, may be created.Fig 1 shows the a graph of γ−sheet of boron clusters for the particular values of p = 6 and q = 3.We established our computational findings in this part for γ−sheet of boron clusters (it is denoted by γ(p, q)).The count of atoms and edges in γ(p, q) are 5pq + 2q and 12pq − p + q respectively, shown in the Table 1.In Table 2, we partitioned the lines based on ev-valency of the γ(p, q).In Table 3, we partition the atoms based on ve-valency of γ(p, q).We partition the lines based on ve-valency of the end atoms of γ(p, q) in Table 4.

Main results
In this part, we will determine different kinds of descriptors' ev-valency and ve-valency based descriptors, which are listed as follows; • ev−valency Zagreb Index.Using the statistics given in Table 2, we determine the ev-valency based Zagreb descriptor of γ(p, q) as: • The first ve-valency Zagreb α descriptor.
Using the statistics given in Table 3, we determine the ve-valency based Zagreb descriptor of γ(p, q) as: • The first ve-valency Zagreb β descriptor. https://doi.org/10.1371/journal.pone.0303570.t004 Using the statistics given in Table 4, we determine the ve-valency based Zagreb descriptor of γ(p, q) as: • The second ve-valency Zagreb descriptor.
Using the statistics given in Table 4, we determine the ve-valency based Zagreb descriptor of γ(p, q) as: ðd ve ðuÞ � d ve ðvÞÞ; • The ve-valency Randić descriptor.
Using the statistics given in Table 4, we determine the ve-valency based Randić descriptor of γ(p, q) as: R ve ðgðp; qÞÞ ¼ X uv2Eðgðp;qÞ ðd ve ðuÞ � d ve ðvÞÞ À 1 2 ; Using the statistics given in Table 2, we determine the ev-valency based Randić descriptor of γ(p, q) as: • The atom-bond connectivity descriptor.
Using the statistics given in Table 4, we determine the ve-valency based atom-bond connectivity descriptor of γ(p, q) as: ABC ve ðgðp; qÞÞ ¼ X   Using the statistics given in Table 4, we determine the ve-valency based geometric arithmetic descriptor of γ(p, q) as: GA ve ðgðp; qÞÞ ¼ X • The sum-connectivity descriptor.
Using the statistics given in Table 4, we determine the ve-valency based sum connectivity descriptor of γ(p, q) as:

Numerical and graphical representation and discussion
Both numerical and graphical calculations are used to determine the ve and ev for ten distinct categories of valency-based topological descriptors for the γ(p, q).The behavior of the first Zagreb alpha descriptor, first Zagreb beta descriptor, and second Zagreb descriptor is practically identical in the growing orientation as the quantity of n rises, while the appreciate of the ev Zagreb descriptor improves very quickly with the increased value of n, as can be seen in  The numerical and graphical representations of γ(p, q) are shown below.

Closing remarks
Topological indicators have several uses in fields such as chemical graph theory, computer science, networks, agriculture, etc.These characteristics aid in determining how their frameworks behave.We generated 10 distinct sorts of topological descriptors based on ev and ve valency for the γ-sheet of boron clusters.Their numerical values for different amounts of n have been computed after their clear formulations have been determined.In addition, we compared the charts and spoke about their behavior.We note that when the quantity of n grows, all descriptor values rise, results are shown in the Tables 5 and 6.

Future directions and open problems
Researchers working in topological numbers of graphs will find guidance and direction in this section.There are many topological numbers, like connection number based topological index is one the next work can be done after this chosen topic.

Limitations of the proposed model
There are few limitations of this work, like Computational Complexity: Certain topological indices or computational models may exhibit significant computational complexity, rendering them unfeasible for handling extensive datasets or real-time applications.

Fig 2 .
The patterns of behavior of the atom bond connectivity descriptor and geometric arithmetic descriptor are hardly noticeably growing with an elevated value of n, but the value of the ev Randić descriptor has a very quick rise with the increased value of n, as can be seen in Fig3.

The γ−sheet of boron clusters
• The atom-bond connectivity descriptor: ABC ve ðGÞ ¼ X uv2EðGÞ ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi d ve ðuÞþd ve ðvÞÀ 2 d ve ðuÞ�d ve ðvÞ q• The geometric-arithmetic descriptor: GA ve ðGÞ ¼

The ev-valency Randić descriptor.
Using the statistics given in Table4, we determine the ve-valency based Harmonic descriptor of γ(p, q) as:

Table 5 . Numerical comparison of M ev
; M ave 1 ; M bve 1 , M ve 2 and R ev .